Have you checked out the web site http://www.shelledwarriors.co.uk/forum/index.php? Lots of good information on keeping torts in the UK, including housing, etc. Seems like there are quite a few Mediterranean torts, due to their cold hardiness.
I am in the Pacific NW of the US, and their site helped me design my outdoor enclosure regarding drainage, etc.
There are no breeds of tortoises. Unlike dogs, cats, horses, etc., turtles are captive wild animals, and not domestic, so they do not come in breeds, only in naturally occurring species, subspecies, and races or regional variants.
To answer your question, though, I would think the most common tortoises to be kept in both Europe and the US are in the Testudo genus, but I am not sure which species or subspecies are the most popular. It seems the Russian is very popular here in America while, based on what I've read from AC Highfield, the Hermann tortoise appears to be very popular in the UK. The Greek seems to be a bit less hardy than these two species, and so seems a bit less widespread, but still quite common in the pet trade.
Im in the uk and I have a horsefield. They do take a lot of looking after so make sure you can make the commitment to them. It's like having a child ha ha